Weakly nonlinear elastic plane waves in a cubic crystal

The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

“…The possibility of the appearance of quadratically nonlinear coupling in the evolution equations for shear elastic waves propagating along the threefold symmetry acoustic axis direction in a cubic crystal was shown for the first time in our paper [3]; see also [4,5]. Experimental confirmation of this fact was announced in [20].…”

“…Each symmetry diminishes the number of elastic stiffness constants. The number of constants for the most symmetric of cubic crystals is reduced to three second-order constants and six third-order constants [5]. We will be interested in the most symmetric of cubic crystals -those belonging to the class m3m.…”

“…This is motivated by the fact that many crystals appearing in nature belong to this class, for example, all metals of cubic symmetry, Au, Fe, Ni, Ag, and ionic crystals such as, for example, NaCl, KCl, LiF, AgCl, and many more. Let us consider a cubic crystal of class m3m in which the strain energy W is defined by three second-order and six third-order elastic constants [5]: c 12 , c 44 , c 111 , c 112 , c 144 , c 123 , c 166 , c 456 ).…”

“…Our aim in this work is to derive equations similar to those in [2] but by a different method. Here, unlike the presentation in [2] (see also [3][4][5][6][7][8]), where the method of weakly nonlinear geometric optics was used, we will apply a double-scale expansion. Instead of introducing a new fast variable and applying geometric optics-type asymptotics, which leads to evolution equations with three independent variables, we introduce here just two new independent variables: a slow time variable τ and a characteristic variable θ .…”

Deformation in micropolar cubic crystal due to various sources

“…Kobayashi and Giga (2001) investigated anisotropy and curvature effects for growing crystals. Domanski and Jablonski (2001) studied resonances of nonlinear elastic waves in cubic crystal. Destrade (2001) considered the explicit secular equation for surface acoustic waves in monoclinic elastic crystals.…”